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2 SUBSTITUTION OF POWER SERIES IN A POWER SERIES 201

(9.1.6) Suppose two power series (0_vz^v) and (b_vz^v) are absolutely summable
and have the same sum in a poly disk P; then a_v = b_vfor every v e W.

Use induction on p; for p = 1, the result follows at once from (9.1.5).
Taking the difference of the two power series, we can assume b_v = 0 for

every v; applying (9.1.4) with q=p-\, we have £ #_n(z')z£ = 0, hence

g_n(z') = 0 for every n and every z' in the projection F of P on K^p-1; the
induction hypothesis applied to each g_n yields then a_v = 0 for every v.

PROBLEMS

1. Let (c_vz^v) be a power series in p variables z, (1 ^i^p); let a = (a_ls ...,0_P)EK^P.
In order that a real number r > 0 be such that, for any / e K such that |/| < r, the series
(c_v(ta_t)ⁿⁱ - • - (ta_p)^np) be absolutely summable, it is necessary and sufficient that

1 ,£

log r + _n logllcvll -f £ n_t log|fl, ^ 0

for all but a finite number of indices v = (n^ . . . , n_p) (apply (9.1.2).)

In particular, for/? = 1, there is a largest number R 5= 0 (the "convergence radius,"
which may be +00) such that the series (c_nzⁿ) is convergent for z\ <R, and that
number is given by 1/R = lim (sup(||c_n+k||^1/<n+fc))), which is also written lim • sup||c_n||^1/fl

n-K» ft>0 JI-+00

(cf. Section 12-7). When in particular lim ||c_n||^1/fl exists, it is equal to 1/R.

n-*oo

2. Give examples of power series in one complex variable, having a radius of convergence
R = 1 (Problem 1) and such that:

(1) the series is normally convergent for |z| = R;

(2) the series is convergent for some z such that |z = R, but not for other points of
that circle;

(3) the series is not convergent at any point of |z| = R.

3. Give an example of a power series in two variables, which is absolutely summable at

— — _t — — I . (Replace z by

ziZ₂ in a power series in one variable.)

4. Let (CflZ"), (d_nzⁿ) be two power series in one variable with scalar coefficients; if their
radii of convergence (Problem 1 ) are R and R', and neither R nor R' is 0, then the
radius of convergence R" of the power series (c_nd_nzⁿ) is at least RR^X (taken equal to
-f oo if R or R^x is + oo). Give an example in which R" > RR^X.

2. SUBSTITUTION OF POWER SERIES IN A POWER SERIES

Let Q be a polydisk of center 0 in K⁴, and suppose the p power series
in q variables (b^u*) with scalar coefficients are absolutely summable
in Q (with ju = (m_l9 ..., m_q)₉ u = (u_i9..., u_q), u» =u^ ... u%<). We write



	2 SUBSTITUTION OF POWER SERIES IN A POWER SERIES 201 (9.1.6) Suppose two power series (0_vz^v) and (b_vz^v) are absolutely summable and have the same sum in a poly disk P; then a_v = b_vfor every v e W. Use induction on p; for p = 1, the result follows at once from (9.1.5). Taking the difference of the two power series, we can assume b_v = 0 for every v; applying (9.1.4) with q=p-\, we have £ #_n(z')z£ = 0, hence g_n(z') = 0 for every n and every z' in the projection F of P on K^p-1; the induction hypothesis applied to each g_n yields then a_v = 0 for every v.

	PROBLEMS 1. Let (c_vz^v) be a power series in p variables z, (1 ^i^p); let a = (a_ls ...,0_P)EK^P. In order that a real number r > 0 be such that, for any / e K such that \|/\| < r, the series (c_v(ta_t)ⁿⁱ - • - (ta_p)^np) be absolutely summable, it is necessary and sufficient that 1 ,£ log r + _n logllcvll -f £ n_t log\|fl, ^ 0

	for all but a finite number of indices v = (n^ . . . , n_p) (apply (9.1.2).) In particular, for/? = 1, there is a largest number R 5= 0 (the "convergence radius," which may be +00) such that the series (c_nzⁿ) is convergent for z\ <R, and that number is given by 1/R = lim (sup(\|\|c_n+k\|\|^1/<n+fc))), which is also written lim • sup\|\|c_n\|\|^1/fl n-K» ft>0 JI-+00 (cf. Section 12-7). When in particular lim \|\|c_n\|\|^1/fl exists, it is equal to 1/R. n-*oo 2. Give examples of power series in one complex variable, having a radius of convergence R = 1 (Problem 1) and such that: (1) the series is normally convergent for \|z\| = R; (2) the series is convergent for some z such that \|z = R, but not for other points of that circle; (3) the series is not convergent at any point of \|z\| = R. 3. Give an example of a power series in two variables, which is absolutely summable at

	— — _t — — I . (Replace z by ziZ₂ in a power series in one variable.) 4. Let (CflZ"), (d_nzⁿ) be two power series in one variable with scalar coefficients; if their radii of convergence (Problem 1 ) are R and R', and neither R nor R' is 0, then the radius of convergence R" of the power series (c_nd_nzⁿ) is at least RR^X (taken equal to -f oo if R or R^x is + oo). Give an example in which R" > RR^X.

	2. SUBSTITUTION OF POWER SERIES IN A POWER SERIES Let Q be a polydisk of center 0 in K⁴, and suppose the p power series in q variables (b^u)* with scalar coefficients are absolutely summable in Q (with ju = (m_l9 ..., m_q)₉ u = (u_i9..., u_q), u» =u^ ... u%<). We write